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<body class="ignore-math">
<h3 class="heading"><span class="type">Paragraph</span></h3>
<ul class="disc">
<li>
<p><dfn class="terminology">Linearity</dfn></p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\cal L}[a f(t)+b g(t)]=aF(s)+bG(s).
\end{equation*}
</div>
<p class="continuation">Examples:</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\mathcal L}[3t+4] = 3\frac{1}{s^2}+4\frac{1}{s}.
\end{equation*}
</div>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\mathcal L}[\cos at + i\sin at]={\mathcal L}[e^{iat}] \  \ {\rm by \ DeMoivre}.
\end{equation*}
</div>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\mathcal L}[e^{iat}] = \frac{1}{s-ia}=\frac{s}{s^2+a^2} + \frac{ia}{s^2+a^2}.
\end{equation*}
</div>
<p class="continuation">Hence, equating real and imaginary parts and using linearity</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\mathcal L}[\cos at]=\frac{s}{s^2+a^2}
\end{equation*}
</div>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\mathcal L}[\sin at]=\frac{a}{s^2+a^2}.
\end{equation*}
</div>
</li>
<li>
<p><dfn class="terminology">Time Shifting</dfn></p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\cal L}[f(t-t_0)]=e^{-t_0s} F(s).
\end{equation*}
</div>
</li>
<li>
<p><dfn class="terminology">Shifting in s-Domain</dfn></p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\cal L}[e^{s_0t}f(t)]=F(s-s_0).
\end{equation*}
</div>
</li>
<li>
<p><dfn class="terminology">Time Scaling</dfn></p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\cal L}[f(at)]=\frac{1}{|a|} F(\frac{s}{a})
\end{equation*}
</div>
</li>
<li>
<p><dfn class="terminology">Convolution</dfn></p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\cal L}[f(t)*g(t)]=F(s) G(s),
\end{equation*}
</div>
<p class="continuation">where <span class="process-math">\(f(t)*g(t)=\int_0^t f(t-\tau) g(\tau) \, \mathrm{d}\,\tau=\int_0^t f(\tau) g(t-\tau) \, \mathrm{d}\,\tau\text{.}\)</span>Example:</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\mathcal L}^{-1}\left[ \frac{f(s)}{s} \right].
\end{equation*}
</div>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\mathcal L}^{-1}[f(s)]=f(t), \ \ {\rm and \ } {\mathcal L}^{-1}\left[\frac{1}{s}\right]=1=g(t),
\end{equation*}
</div>
<p class="continuation">so</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\mathcal L}^{-1}\left[ \frac{f(s)}{s} \right]=\int_0^t f(\theta) \, d\theta.
\end{equation*}
</div>
</li>
<li>
<p><dfn class="terminology">Differentiation in Time Domain</dfn></p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\cal L}\left[\frac{d}{dt}f(t)\right]=sF(s)-f(0).
\end{equation*}
</div>
<p class="continuation">In general, we have</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\cal L}\left[ \frac{d^n}{dt^n}f(t)\right]=s^n F(s)-s^{n-1} f(0)-\cdots-s f^{(n-2)}(0)-f^{(n-1)}(0).
\end{equation*}
</div>
</li>
<li>
<p><dfn class="terminology">Differentiation in s-Domain</dfn></p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\cal L}[t f(t)]=-\frac{d}{ds}F(s).
\end{equation*}
</div>
<p class="continuation">This can be proven by differentiating the Laplace transform:</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\frac{d}{ds}F(s)=\int_{0}^\infty f(t) \frac{d}{ds} e^{-st} dt
=\int_{0}^\infty (-t) f(t) e^{-st} dt
\end{equation*}
</div>
<p class="continuation">Repeat this process we get</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\cal L}[t^n f(t)]=(-1)^n\frac{d^n}{ds^n}F(s).
\end{equation*}
</div>
</li>
<li>
<p><dfn class="terminology">Integration in Time Domain</dfn></p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\cal L}\left[\int_{0}^t f(\tau) d\tau \right]=\frac{F(s)}{s}.
\end{equation*}
</div>
<p class="continuation">This can be proven by realizing that</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
f(t)*u_0(t)=\int_{0}^t f(\tau) u_0(t-\tau) d\tau    
=\int_{0}^t f(\tau) d\tau
\end{equation*}
</div>
<p class="continuation">and therefore by convolution property we have</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\cal L}[f(t)*u_0(t)]=F(s)\frac{1}{s}   .
\end{equation*}
</div>
<p class="continuation">Note <span class="process-math">\(u_0(t)=1\)</span> and <span class="process-math">\({\cal L}[u_0(t)]=1/s\text{.}\)</span></p>
</li>
</ul>
<span class="incontext"><a href="sec8_2.html#p-439" class="internal">in-context</a></span>
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